Abstract

Entanglement in bipartite continuous-variable systems is investigated in the presence of
partial losses, such as those introduced by a realistic quantum communication channel, e.g. by
propagation in an optical fiber. We find that entanglement can vanish completely for partial
losses, in a situation reminiscent of so-called entanglement sudden death. Even states with
extreme squeezing may become separable after propagation in lossy channels. Having
in mind the potential applications of such entangled light beams to optical communications,
we investigate the conditions under which entanglement can survive for all partial losses.
Different loss scenarios are examined and we derive criteria to test the robustness of
entangled states. These criteria are necessary and sufficient for Gaussian states. Our
study provides a framework to investigate the robustness of continuous-variable
entanglement in more complex multipartite systems.

pacs:

03.67.Bg,03.67.Pp, 42.50.Xa, 42.50.Dv

pacs:

The dynamics of open quantum systems leads in general to a degradation of
key quantum features, such as coherence and entanglement. Since entanglement
is considered to be an important resource for applications in quantum information,
its degradation may seriously hinder the envisioned protocols. Careful analyses of
environment-induced loss of entanglement are thus important steps in quantum
information science. In the discrete-variable scenario, studies of 2-qubit systems
have shown that entanglement can be completely lost after a finite time of interaction
with the environment, an effect now mostly known as Entanglement Sudden Death
(ESD) (1); (2). Quantum information can also be conveyed, stored,
and processed by continuous-variable (CV) systems. Bright beams of light can be
described by means of CV field quadratures and are natural conveyors of quantum
information. Unavoidable transmission loss is the fiercest enemy for quantum
communications. It has recently been observed that losses may lead to complete
disentanglement in Gaussian CV systems (3); (4).
This phenomenon is a partial-loss analog of the finite-time disentanglement observed
in qubit systems.

The simplest CV systems one can consider are those described by Gaussian statistics.
Gaussian states are indeed well studied (5) and fairly well
characterized. For instance, there exist necessary and sufficient criteria for Gaussian-state
entanglement of up to 1×N systems (in which one subsystem is collectively
entangled to N other subsystems) (6); (7). In spite of all this
knowledge, the sensitivity of entanglement to the interaction with the environment is
still not completely mapped. As experimentally observed by Coelho et al.(3)
and by Barbosa et al.(4), some Gaussian states become separable for
partial losses while others remain entangled. What distinguishes one class of states from
the other? Are there only two classes of such states? Is it sufficient to produce states with
a large degree of squeezing in order to avoid disentanglement? Is there any strategy
involving local operations to protect states against disentanglement?

In this paper, we extend the treatment of ref. (4) and provide answers to
some of these questions. We theoretically analyze the conditions leading to CV disentanglement
in the simplest case of bipartite systems. In the framework of open system dynamics, the effect
of a lossy channel (without any added noise) is equivalent to the interaction with a reservoir at zero
temperature. The property of entanglement resilience to losses will be referred to as ‘robustness’.
Entanglement robustness is assessed by entanglement criteria previously derived by other authors.
For general CV states, these criteria provide sufficient conditions for the robustness of bipartite
systems. Necessary and sufficient entanglement criteria for Gaussian states lead to necessary
and sufficient conditions for entanglement robustness upon propagation in lossy channels.
Entanglement of CV Gaussian states may be created by a number of different strategies
such as, for instance, passive operations on initially squeezed states (8).
We shall not discuss these in detail here, but take for granted initially entangled states.

A thorough investigation reveals the possibility of distinct entanglement dynamics as
losses are imposed on the subsystems. We consider realistic scenarios, as depicted
in Fig. 1. A bipartite entangled state is the quantum resource
of interest. It can be distributed to two parties who wish to communicate, as in
Fig. 1 (a), in a scenario which we refer to as a dual-channel
communication scheme. Another possibility would be that one of the parties holds
the quantum state generator and only one mode needs to propagate through a
lossy quantum channel, as in Fig. 1 (b). We refer to this
situation as a single-channel scheme. One could surmise, in principle, that it is equivalent
to concentrate losses in a single channel or split them among two channels. If
our channels are optical fibers, losses increase exponentially with the propagation
distance. Thus, one could think that propagation in a single fiber over a certain distance
would have the same effect as propagation of both modes, each in one fiber, over half
the distance (which would result in the same overall losses). This is not correct: for certain states,
one could propagate one of the modes over an infinite distance in a single
lossy channel without losing entanglement, whereas entanglement would disappear after
a finite propagation distance if both modes were to suffer losses.

Figure 1: (Color online) (a) Dual-Channel Losses: An entangled quantum state is distributed to
two parties, Alice and Bob, over two lossy quantum channels; (b) Single-Channel
Losses: Alice holds the quantum state generator and only distributes
one entangled mode to Bob over a single lossy quantum channel.

These different scenarios lead to the introduction of a formal classification, consisting
of three robustness classes. On one extreme, the entanglement of fully robust states vanishes
only for total attenuation of either beam. On the opposite extreme, fragile states become
separable for partial attenuations on either beam or a combination of both. An intermediate
class of partially robust states shows either robustness or fragility depending on the way
losses are introduced. Thus, imposing losses on one field may be less harmful in a quantum
communication system than distributing both beams over two lossy channels. Furthermore,
we show that even states with very strong squeezing (e.g. amplitude difference squeezing,
as in twin beams produced by an above-threshold OPO) can disentangle for partial losses.
A moderate excess noise, commonly encountered in existing experiments, suffices for this.
In addition, one could speculate that pure states would necessarily be robust. We provide
an example of a pure state that disentangles for partial losses as well.

The paper is organized as follows. In Section II we establish notation and the
basic reservoir model (the environment). In Section III a sufficient criterion to
determine the robustness of the entangled state is demonstrated. In Section IV
we extend the robustness criterion, resulting in a necessary and sufficient robustness
condition for all Gaussian bipartite states. The different classes of entanglement robustness
against losses in each channel are defined in Section V. In Section VI
we examine particular quantum states commonly treated in the literature. A final Section VII
is focused on the main physical results and implications of our findings.

The quantum properties of Gaussian states are completely
characterized by the second order moments of the appropriate observables.
The choice of observables depends on the system under consideration.
In the case of the electromagnetic field, a complete description can be
given in terms of orthogonal field quadratures. We will consider
the amplitude and phase quadratures, respectively
written as ^pj=(^a†j+^aj) and
^qj=i(^a†j−^aj) in terms of the
field annihilation ^aj and creation ^a†j
operators. The indices j=1,2 stand for the two field modes of our bipartite system.
The quadrature operators obey the commutation relation [^pj,^qj]=2i,
from which we obtain an uncertainty product lower bound of one. The standard
quantum level (SQL) is thus equal to one, representing the noise power present in
the quadrature fluctuations of a coherent state.

It is useful to organize the second order moments in the form of a 4×4 covariance matrix V.
Its entries are the averages of the symmetric products of quadrature fluctuation operators

V=12⟨δ^ξδ^ξT+(δ^ξδ^ξT)T⟩,

(1)

where ^ξ=(^q1,^p1,^q2,^p2)T is the column vector of quadrature
operators, and δ^ξ=^ξ−⟨^ξ⟩ are the fluctuation operators with
zero average. Similar notation will be valid for the individual quadratures, e.g. δ^p1.
The noise power is proportional to the variance of the fluctuation,
denoted for a given quadrature by (e.g.) Δ2^p1=⟨(δ^p1)2⟩.
The Heisenberg uncertainty relation can be expressed as (9); (6)

V+iΩ

≥

0,

(2)

whereΩ=[J00J],

and

J=[01−10].

The covariance matrix can be divided in three 2×2 submatrices,
from which two (Aj) represent the reduced covariance matrices of the individual subsystems
and one (C) expresses the correlations between the subsystems

V=(A1CCTA2).

(3)

The correlations originate from both classical and quantum backgrounds,
and cannot be directly associated to entanglement without considering the
properties of each subsystem. As we will see, the occurrence of ESD is
related to the presence of uncorrelated noise in the system, normally in
the form of unbalanced or insufficient correlations between different subsystems or quadratures.

For bipartite Gaussian states, there exist necessary and sufficient entanglement
criteria (6); (10). These criteria are the basis for our assessment of
entanglement robustness.

First, we need to adopt a model for the quantum channel. Here
we consider the realistic case of a lossy bosonic channel, equivalent to
the attenuation of light by random scattering.
Losses are modeled by independent beam splitters placed in the beam paths.
Each beam splitter transformation combines one field mode with the vacuum field.
In the absence of added noise, it can be associated to a reservoir at zero
temperature.

A Gaussian attenuation channel transforms the field operators according to (11); (12)

^aj⟶^a′j=√Tj^aj+√1−Tj^a(E)j,

(4)

where Tj is the beam splitter transmittance and ^a(E)j
is the annihilation operator from the environment. It acts
on the covariance matrix as

V′=L(V)=L(V−I)L+I,

(5)

where L=diag(√T1,√T1,√T2,√T2) is the loss matrix
and I is the 4×4 identity matrix.

The question we address here regards the behavior of entanglement as the covariance
matrix undergoes the transformation of Eq. (5).

We direct our attention, in a first moment, to the entanglement criterion presented
in Ref. (10), here referred to as the Duan criterion.
According to them, a sufficient condition for the existence
of entanglement is obtained by fulfilling the inequality

WD=Δ2^u+Δ2^v−(a2+1a2)<0,

(6)

where

^u=1√2(|a|^p1−1a^p2)and^v=1√2(|a|^q1+1a^q2).

(7)

The ^pi and ^qi are quadrature operators, obeying the commutation
relations stated above and a is an arbitrary real nonzero number. The quadrature
combinations ^u and ^v are collective operators corresponding to
the original example of Einstein, Podolsky and Rosen (EPR) (13). As such,
they are called EPR-like collective operators.

The quantity WD can be viewed as an entanglement witness. We shall use the
symbol ’W’ for witnesses in general. The presence of a given property is signaled by a
negative value of the corresponding witness. As a merely sufficient criterion, no statement
can be made if WD≥0: the state could be either separable or entangled. Nevertheless,
the witness WD is compelling from a practical point of view because it does not require
full knowledge of the covariance matrix, simplifying the detection of entanglement in experiments.
The downside is its limited detection ability.

For a=1, entanglement can be detected by a balanced beam
splitter transformation of the input fields followed by a measurement of squeezing in the
two output fields (14); (15). Alternatively, one can measure the quadrature variances
Δ2^pi and Δ2^qi of each field and the cross correlations
cp=⟨δ^p1δ^p2⟩ and cq=⟨δ^q1δ^q2⟩.
The optimum choice for the parameter a that minimizes WD is
a2=√σ2/σ1, where the σj are given by

σj=Δ2^pj+Δ2^qj−2=trAj−2.

(8)

The sign indeterminacy in a is solved by taking into account the signs of the quadrature
correlations. With these considerations, one arrives at the minimized form of the Duan criterion

WM=σ1σ2−(cp−cq)2<0.

(9)

Eq. (9) provides the first insight into the robustness of bipartite states.
The crucial fact to be observed is that the sign of WM is conserved by attenuations.
In fact, using Eq. (5), the correlations transform as c′p=√T1T2cp
and c′q=√T1T2cq, while σ′j=Tjσj.
The attenuation operation factorizes in the entanglement witness,

W′M=T1T2WM.

(10)

Therefore, an initially entangled state satisfying Eq. (9) will not disentangle
under partial losses. This fact was experimentally verified by Bowen
et al.(16).

Entangled states satisfying the Duan criterion do not disentangle for partial losses imposed on any mode:
they are fully robust. Among them lie the two-mode squeezed states, a large class of states for
which both EPR-like observables are squeezed (17); (15); (18).

Since WM is only a sufficient witness, the existence of robust states
for which WM≥0 cannot be excluded. Below, we demonstrate a necessary and sufficient
criterion for robustness of Gaussian states, effectively determining the boundary
between robust and fragile states.

In order to obtain clear-cut conditions for the robustness of entanglement,
we must employ a necessary and sufficient entanglement criterion.
By analyzing whether the subsystems remain entangled or become separable
upon attenuation, we will classify all bipartite Gaussian states.

iv.1 The PPT Criterion

We find a convenient separability criterion in the requirement of positivity under partial
transposition (PPT) of the density matrix for separable states (19); (20). An
entangled state, on the other hand, will necessarily lead to a negative partially transposed
density matrix, which is non-physical.

The partial transposition (PT) of the density operator is equivalent in the level of the Wigner function to the operation
of time-reversal applied to a single subsystem. On the covariance matrix level, time-reversal
is obtained by changing the sign of the momentum (for harmonic oscillators), or the sign
of the phase quadrature of one mode (for electromagnetic fields), in this manner affecting the
sign of its correlations (6).

Physical validity is assessed using Eq. (2). The uncertainty relation can be
recast into a more explicit form by expressing it in terms of the determinants of the covariance
matrix and its submatrices as

1+detV−2detC−∑i=1,2detAj≥0.

(11)

The PT operation modifies the sign of detC, resulting in the following condition for entanglement (6)

Wppt=1+detV+2detC−∑i=1,2detAj<0.

(12)

Since all separable states fulfill Wppt≥0, Wppt is a sufficient entanglement
witness. For Gaussian states, it is a necessary witness as well, and the equation Wppt=0
traces a clear boundary in the space of bipartite Gaussian states, setting apart the subspaces of
separable and entangled states.

It is convenient to recall here that the purities of Gaussian
states are directly related to the determinant of the covariance
matrices (21)

μ

=

(detV)−12,

(13)

μj

=

(detAj)−12,

(14)

so that the entanglement witness of Eq. (12) involves the total purity of the systems,
the purity of each subsystem, and the shared correlations.

iv.2 Covariance Matrix under Attenuation

Applying the witness of Eq. (12) to the attenuated covariance matrix of Eq. (5), one obtains

W′ppt(T1,T2)=1+detV′+2detC′−∑j=1,2det(A′i),

(15)

from which W′ppt(T1=1,T2=1)=Wppt. From Eq. (5), it follows that the individual
submatrices transform as C′=√T1T2C and A′j=Tj(Aj−I)+I under attenuations.
The bilinear dependence of Eq. (9) on T1 and T2 which led to a constant sign of the
witness is not expected here and robustness is not a general feature of bipartite entangled states.

In Appendix A, we derive an explicit transmittance-dependent form of W′ppt(T1,T2). We can factor
out a term T1T2, which cannot change the sign of Wppt. It assumes the form

W′ppt(T1,T2)=T1T2WR(T1,T2).

(16)

The reduced witness WR preserves the sign of W′ppt (except for T1=T2=0, for which
we know both modes are in their vacuum states and W′ppt=0), maintaining only the relevant
dependence on T1 and T2. It reads

WR(T1,T2)=T1T2Γ22+T2Γ12+T1Γ21+Γ11.

(17)

The expressions for the coefficients Γij in terms of the covariance matrix entries are given
in Appendix A. We note that they are regarded as constants here, independent of T1 and T2.

Figure 2: (Color online) Possible behaviors of the PPT entanglement witness W′ppt under attenuation, as a
function of the transmittances T1 and T2. (a) Fully robust entanglement. (b) Fragility for any combination of beam attenuations. (c) Separable state. (d) Single-channel partial robustness -
either mode: the state is robust for any individual attenuation, but not for a combination of attenuations,
such as equal attenuations. (e) Single-channel partial robustness - specific mode, i.e., the state is
robust when one mode is attenuated but presents ESD upon attenuation of the other mode.

The different dynamics of entanglement under losses appear in the witnesses W′ppt and WR.
Fig. 2 depicts four entangled states (three of them fragile) plus a separable state
under attenuation. The plots show W′ppt(T1,T2) based on the covariance matrix

V=⎡⎢
⎢
⎢
⎢
⎢⎣Δ2q10cq00Δ2p10cpcq0Δ2q200cp0Δ2p2⎤⎥
⎥
⎥
⎥
⎥⎦,

(18)

constructed from diagonal submatrices. This simple form of V, observed in the experiments of Ref. (4), suffices to span all types of entanglement
dynamics of Gaussian states.

As the correlation cq is varied, different types of entanglement dynamics are observed. Modifying this parameter while keeping constant the other entries of the covariance matrix is equivalent to adding uncorrelated noise to the system (for instance, classical phonon noise dependent on the temperature of the non-linear crystal (4); (22)).
In Fig. 2a (cq=1.275), a state violating the Duan criterion is fully robust, as expected.
Disentanglement does not occur for finite losses imposed on any of the fields. In Fig. 2b,
the choice cq=0.893 characterizes a state for which ESD occurs for partial attenuation in a single channel
(mode) or in both channels. This represents the most fragile class of states. In Fig. 2c (cq=0.3825),
the initial state is separable and it naturally remains separable throughout the whole region of attenuations.

A more subtle entanglement dynamics appears in Fig. 2d (cq=1.033). The state is robust
against any single-channel attenuation but may become separable if both modes are attenuated. Such a
state would suffice as a resource for quantum communications involving single-channel losses.

If we consider a more general covariance matrix, with asymmetric modes, the system may be robust against
losses on one mode, but not on the other. This is observed in Fig. 2e, where W′ppt is
calculated for the covariance matrix

V=⎡⎢
⎢
⎢⎣2.5500.653001.800−0.7970.65301.6200−0.79701.32⎤⎥
⎥
⎥⎦.

(20)

This particular covariance matrix is obtained from Eq. (19), with cq=1.033, by imposing
the attenuation T2=0.40. Before this attenuation, the state was partially robust, as in Fig. 2d.
It remains robust against losses on mode 2, but now disentanglement with respect to losses solely on mode 1
may occur. This illustrates the fact that the new states produced upon attenuation become
more fragile. Since attenuation is a Gaussian operation, states cannot become more robust upon attenuation (23); (24).

iv.3 Full Robustness

We show here that fully robust states can be directly identified from the covariance matrix.
In order to obtain the necessary condition, we note from Eq. (17) that the entanglement dynamics
close to complete attenuation is dominated by Γ11. Thus, an initially entangled state WR(T1=1,T2=1)<0
with Γ11>0, must become separable for sufficiently large attenuation, from which we derive the witness

Wfull=Γ11=σ1σ2−tr(CTC)+2detC.

(21)

Wfull≤0, provided Wppt<0, supplies a simple, direct, and general condition for testing the entanglement robustness of
bipartite Gaussian states.

The robustness cannot depend on the choice of local measurement basis for each mode since, as discussed in
Appendix A, local rotations commute with the operation of losses. In other words, local passive operations, such as
rotations and phase shifts, do not change the robustness. By using local rotations
to diagonalize the correlation matrix C, we obtain

W(D)full=σ1σ2−(cp−cq)2≤0,

(22)

which coincides with WM, of Eq. (9). Thus, the Duan criterion in the simple form of Eq. (9)
is a particular case of Eq. (21) when the correlation submatrix is diagonal. For Gaussian states given by
covariance matrices with diagonal correlation submatrix, WM is a necessary and sufficient witness for robust
entanglement, but only sufficient otherwise.

iv.4 Partial Robustness

As seen in Fig. 2, there exist states which can be robust against single-channel
losses, yet disentangle for finite losses split among two channels. Similar to the procedure in the
previous section, we will define witnesses capable of identifying partial robustness.

Let us consider the case T2=1 for definiteness. The attenuated witness of Eq. (17) becomes

WR(T1,T2=1)=(Wppt−W1)T1+W1,

(23)

where

W1=Wfull+Γ21

(24)

(see Appendix A for the expression of Γ21). The analysis of W1 follows the same lines
used in the case of fully robust states, with the simplification that the witness depends linearly on the
attenuation. Thus, there is only one possible path cutting the plane WR(T1,T2=1)=0.
The fraction of transmitted light for which ESD occurs is

Tc1

=

W1W1−Wppt.

(25)

From Wppt<0, it follows that 0<W1<W1−Wppt, to assure that Tc1 exists as a
meaningful physical quantity (0<Tc1<1) whenever W1>0.

Therefore, an entangled state satisfying W1≤0 is robust against losses in channel 1, and
W1 is the witness for this type of robustness. The corresponding analysis regarding attenuations
on the subsystem 2 yields the witness

W2=Wfull+Γ12,

(26)

with the same properties of W1. A relation analogous to Eq. (25) holds for Tc2.
Both witnesses are invariant under local rotations, as expected.

Based on the different dynamics of entanglement of Fig. (2), we
propose a classification of bipartite entangled states according to their resilience
to losses. We take guidance in the sign of the reduced witness WR(T1,T2), which
is a hyperbolic paraboloid surface. The contour defined by the condition WR(T1,T2)=0
provides a complete description of the entanglement dynamics in terms of Γij. As depicted
in Fig. 2, there are three relevant situations. Bipartite entangled Gaussian states can be
assigned to the following different classes:

Fully robust states remain entangled for any partial attenuation: WR(T1,T2)<0,∀T1,2.

Partially robust states: (a) symmetric – remain entangled against losses on a single mode, but may
disentangle for combinations of partial attenuations on both modes:
WR(T1,T2=1)<0,∀T1, and WR(T1=1,T2)<0,∀T2.
(b) asymmetric – remain entangled against losses on a specific mode, but may disentangle for partial losses
on the other mode: either WR(T1,T2=1)<0,∀T1, or WR(T1=1,T2)<0,∀T2.

Fragile states disentangle for partial attenuation on any mode or combinations of partial attenuations on both modes.

For a complete classification of all bipartite Gaussian states, one should include the separable states.

With the witnesses previously defined, we have necessary criteria to assess the robustness of all bipartite
Gaussian states. A state will be robust with respect to losses imposed on subsystem 1 if

W1≤0.

(27)

Likewise, robustness to losses on subsystem 2 is given by

W2≤0.

(28)

States will be partially robust if at least one of W1 or W2 is negative or even if both
are negative simultaneously (partially robust – symmetric). Only if WR(T1,T2)<0,∀T1,2
will we have full robustness.

As mentioned above, this classification is of practical interest. Several quantum communication
protocols using continuous variables can be realized by one of the parties (Alice) locally producing
the entangled state and sending only one mode to a remote location. The other party (Bob) then
performs operations on his part of the state, according to instructions sent by Alice through a
classical channel. The success of these communication schemes strongly depends on the
losses that the subsystem of Bob may undergo, which could be detected by an eavesdropper (Eve). In
this situation, Alice must produce entangled states that are at least partially robust in order to avoid problems
with signal degradation. It may not be necessary for her to produce fully robust states: partially robust entangled
states may suffice for successful quantum communication protocols.

In the preceding analysis we have found precise conditions to determine
whether or not bipartite Gaussian entangled states are robust against losses.
Given the practical interest of such states as resources for quantum communication
protocols, we examine here particular Gaussian states that fall within the
classification scheme proposed above. One might think that it should suffice to
generate pure states with a large amount squeezing in order to have robust
entanglement. We begin by providing a specific example of a pure strongly
squeezed state, which is only partially robust. We then examine different forms
of the covariance matrix, in order to map out the different possibilities.

vi.1 Pure and highly squeezed states with only partial robustness

In most experiments, Gaussian bipartite entanglement is witnessed by a
violation of the simplified Duan inequality of Eq. (6). Typically,
this is done by combining highly squeezed individual modes on a beam splitter.
This method allows the creation of arbitrarily strong entanglement in the sense
that quantum information protocols such as teleportation could in principle be
realized with perfect fidelity in the limit of an EPR state.

If such a state is contaminated by uncorrelated classical noise (e.g. from
Brillouin scattering in an optical fiber (27)), it may then become subject to
disentanglement from losses. Even states which are pure may be subject
to disentanglement in a dual-channel scenario. We present below the
covariance matrix for a pure state with these characteristics:

V=⎛⎜
⎜
⎜⎝52.50−47.5000.10500.095−47.5052.5000.09500.105⎞⎟
⎟
⎟⎠.

(29)

This state has a very small symplectic eigenvalue, indicating very
strong entanglement (25). As can be observed in Fig. 3, the state is
partially robust: losses on any single channel do not lead to disentanglement,
while ESD will occur for combined losses in both channels.

Figure 3: (Color online) Entanglement as function of losses for the covariance matrix given by
Eq. (29). Disentanglement may occur only for combined losses on
both modes. In this example, the symplectic eigenvalue (6) is only 0.22 for the initial
state.

Let us now examine different symmetries of the covariance matrix and their
implications on the entanglement dynamics.

vi.2 Symmetric Modes and Quadratures - Fully Robust States

We begin by examining completely symmetric modes, for which
Δ2^p1=Δ2^q1=Δ2^p2=Δ2^q2=s
and ⟨δ^p1δ^p2⟩=⟨δ^q1δ^q2⟩=c,
and ⟨δ^pjδ^qj′⟩=0. The covariance matrix has the form

V=⎛⎜
⎜
⎜⎝s0c00s0−cc0s00−c0s⎞⎟
⎟
⎟⎠.

(30)

Such states can be generated, for instance, by the interference of (symmetric) squeezed states on a
balanced beam splitter (entangled squeezed states) (17); (15). In
this case one has s=νcosh2r and c=νsinh2r, where r is the squeezing parameter and
ν≥1 accounts for an eventual thermal mixedness, representing a correlated classical noise
between the systems.

Entanglement and robustness witnesses are thus

Wppt=(s2−c2+1)2−4s2

(31)

and

Wfull=4[(s−1)2−c2]=4(s2−c2+1−2s),

(32)

from which one directly sees that Wppt<0 and Wfull<0 lead to the same condition (s−1−|c|<0).
Therefore, entangled states with symmetry between the two modes and the two quadratures are fully
robust. The lack of ESD in these systems indicates that strong symmetries lead to entanglement robustness,
even when classical noise is present, as long as it is correlated.

The highly symmetric covariance matrices of Eq. (30) are a particular case of the standard form II of
Ref. (10). For these, the Duan criterion is equivalent to the PPT criterion, which then entails full
robustness for all entangled states with covariance matrices in standard form II. Moreover, since any state
can be brought to standard form II by local squeezing and quadrature rotations without changing its
entanglement (10), any fragile state can be made robust by suitable local unitary operations. The
converse is also true: local squeezing can transform robust states into fragile ones without changing the
entanglement. For instance, if one applies a gate that makes use of local squeezing to a given robust
entangled state, it can become fragile and undergo disentanglement upon transmission. Local squeezing
is one of the important steps in an implementation of a C-NOT (or QND) gate with continuous
variables (26).

vi.3 Symmetric Modes and Asymmetric Quadratures

More general covariance matrices are necessary in order to observe disentanglement. States which are
symmetric on both modes but asymmetric on the quantum statistics of the quadratures have been recently
observed to present ESD (4). The system under investigation consisted of the twin light
beams produced by an optical parametric oscillator, described by a covariance matrix of the form

V=⎛⎜
⎜
⎜
⎜⎝Δ2q0cq00Δ2p0cpcq0Δ2q00cp0Δ2p⎞⎟
⎟
⎟
⎟⎠.

(33)

The entanglement and robustness witnesses read

Wppt=

[(Δ2p)2−c2p][(Δ2q)2−c2q]

(34)

−2Δ2pΔ2q+2cpcq+1

and

Wfull=(Δ2p+Δ2q−2)2−(cq−cp)2.

(35)

In this situation, the subsystems have equal purities (μS=1/√Δ2pΔ2q). The quadrature
variances and correlations are constrained by (Δ2p)2−c2p≥0 and
(Δ2q)2−c2q≥0. We introduce the normalized correlations ¯Cp=cp/Δ2p
and ¯Cq=cq/Δ2q for simplicity. They are bounded by −1≤¯Cj≤1.
These parameters suffice to describe any state with the form of Eq. (33).

Figure 4: (Color online) The space of states with covariance matrices of the form of Eq. (33) is plotted as
a function of the normalized correlations ¯Cp and ¯Cq. Separable states lie in the region IV
; fully robust states are comprised within the region I; partially robust states are in the region II,
and fragile states are in the region III. Points outside of these regions do not correspond to physical states.
Here we use Δ2p=1.80 and Δ2q=2.55. The points included represent the states in Fig. 2a–d.

In Fig. 4 the robustness condition is mapped in terms of the correlations for a fixed purity
μS=0.626, showing the regions corresponding to different robustness classes. Fully robust state (a)
falls within the I region in Fig. 4, while the separable state (c) is located in the IV
region. Within the intermediate region, two different types of fragile states are present. State (d) is partially robust,
lying close to the boundary to robust states. State (b) shows ESD for partial losses in general, lying close to the
boundary to separable states.

Alternatively, following the treatment described in Ref. (4), the covariance matrix
of Eq. (33) can be parametrized in terms of the physically familiar EPR-type operators,

^p±=1√2(^p1±^p2)

(36)

and

^q±=1√2(^q1±^q2).

(37)

Entanglement can be directly observed from the product of squeezed variances of the proper pair
of EPR operators, (^p−,^q+) or (^p+,^q−). Additionally, the entanglement and robustness criteria of symmetric two-mode systems of Eqs. (34)–(35)
can be written in the simpler forms,

Wppt

=

Wprod¯¯¯¯¯¯Wprod,

(38)

Wfull

=

Wsum¯¯¯¯¯¯Wsum,

(39)

where

Wsum

=

Δ2^p−+Δ2^q+−2,

¯¯¯¯¯¯Wsum

=

Δ2^p++Δ2^q−−2,

Wprod

=

Δ2^p−Δ2^q+−1,

¯¯¯¯¯¯Wprod

=

Δ2^p+Δ2^q−−1.

The distinction between robust and partially robust entanglement is clearly illustrated with symmetric modes.
Considering attenuation solely on mode 1 (entirely equivalent to attenuation on mode 2, given the
symmetry), the condition for partial robustness of Eq. (27) yields

W1=Wsum¯¯¯¯¯¯Wprod+Wprod¯¯¯¯¯¯Wsum.

(40)

The condition W1=0 defines the border between partial robustness and fragility. Since a state must
be initially entangled in order to disentangle, obviously

Wfull<0⟹Wppt<0.

(41)

Given the commutation relations between ^p and ^q, Wprod and ¯¯¯¯¯¯Wprod
(or Wsum and ¯¯¯¯¯¯Wsum) cannot be simultaneously negative.
In this context, the condition of Eq. (41) can be restated as

Wsum<0

⟹

Wprod<0or

¯¯¯¯¯¯Wsum<0

⟹

¯¯¯¯¯¯Wprod<0.

(42)

For W1=0,

Wsum¯¯¯¯¯¯Wprod=−Wprod¯¯¯¯¯¯Wsum.

(43)

This equation holds only if Wprod<0 and Wsum>0 (or ¯¯¯¯¯¯Wprod<0 and ¯¯¯¯¯¯Wsum>0).
Thus W1=0 lies between the curves Wppt=0 and Wfull=0.

A plot of the state space in terms of these EPR variables is presented in Fig. 5. Fixed
values for the partial purities, μ+=1/√Δ2^p+Δ2^q+ and
μ−=1/√Δ2^p−Δ2^q−, are assumed, so that we can write the entanglement and
robustness conditions in terms of Δ2^p− and Δ2^q+. The observation of ESD reported
in Ref. (4) was obtained for partially robust states lying in the region delimited by the conditions
Wsum>0 and W1<0.

Figure 5: (Color online) The space of symmetric two-mode states is plotted as a function of the EPR variances
Δ2^q+ and Δ2^p−, normalized to the standard quantum limit (SQL). Separable states lie in the region IV; fully robust entangled states
are within the region I; partially robust states are in the region II and fragile states are in the region
III. The partial purities are μ−=0.7267 and μ+=0.4529.

vi.4 System in Standard Form I

The last case we consider is a covariance matrix in the standard form I (6); (10).
It represents two different modes with symmetric quadratures,

V=⎛⎜
⎜
⎜
⎜⎝s0cq00s0cpcq0t00cp0t⎞⎟
⎟
⎟
⎟⎠.

(44)

The entanglement and full robustness witnesses read

Wppt=(st−c2q)(st−c2p)−s2−t2+2cqcp+1

(45)

and

Wfull=4(s−1)(t−1)−(cq−cp)2.

(46)

The subsystems have purities μ1=s−1 and μ2=t−1. We define the normalized
correlations ¯cj=cj/√st=cj√μ1μ2 as before.

A covariance matrix in standard form I also presents ESD for certain parameters,
spanning all three classes of states described above. Owing to the symmetry in the covariance matrix, ESD in such a system does not occur
for symmetric correlations, ¯cq=−¯cp, independently
of the purities μ1 and μ2.

We have addressed in this paper the issue of entanglement in the open-system dynamics of
continuous-variable (CV) systems. Entanglement is a crucial albeit fragile resource for quantum information
protocols. Understanding its behavior in open systems is very important for future practical applications.

Our analysis is carried out for the simplest possible situation in the CV setting: bipartite Gaussian states
under linear losses. The general study undertaken here
was motivated by the experimental results presented in (3); (4).

Starting from necessary and sufficient entanglement criteria, we
derived necessary and sufficient robustness criteria, which enable us to classify these states
with respect to their entanglement resilience under losses. Having in mind realistic communications
scenarios, we present a robustness classification: states may be fully robust, partially robust, or fragile.
For instance, if one generates an entangled
state for which only one mode will propagate in a lossy quantum channel (single-channel losses),
the conditions derived for partially robust states apply. Such partial robustness would be the minimum
resource required for single-channel robust quantum communications.

On the other extreme, EPR states, for which quantum correlations appear in collective operators of
both quadratures, are the best desirable quantum resource. Their entanglement is resilient to
any combination of losses acting on both modes, only disappearing when the state suffers total loss.
However, a rather likely deviation from such states could already be catastrophic for entanglement: if
a moderate amount of uncorrelated noise (e.g. thermal noise) is introduced in the EPR-type collective operators for one
quadrature, even when the other quadrature remains untouched and is perfectly squeezed, entanglement
can be lost for partial attenuation. This offers a clue to the main ingredients leading to ESD in
bipartite Gaussian states. An appealing example is given by the OPO operating above threshold.
The usual theoretical analysis leads to symmetric modes, with asymmetric quadratures, but no
uncorrelated classical noise. Thus, the OPO is predicted to generate fully robust entangled states.
However, uncorrelated thermal noise originating in the non-linear crystal couples into the two modes (22),
leading to ESD (4).

We have also found that such noise does not necessarily have to imply mixedness.
Even for pure states, the lack of correlation between modes increases the state’s fragility. Robustness
is thus achieved not only for high levels of entanglement between CV systems, but also symmetry
in the form of quantum correlations is desirable. This point was illustrated by our study of mathematical
examples of Gaussian states, for which symmetry implied robustness in spite of mixedness.
We also point out that robustness can be obtained, in principle, for any entangled state by local unitary
operations, such as squeezing and quadrature rotations. However, these operations are not
always simple to implement in an experiment.

As an outlook, we should keep in mind that scalability is one of the main goals in
quantum information research at present. As larger and more complex systems are
envisioned for the implementation of useful protocols, higher orders of entanglement
will be required. Disentanglement for partial losses was experimentally observed in
the context of a tripartite system (3). An understanding of entanglement
resilience for higher-order systems will be important. The methods and analyses developed
here constitute the starting point for such investigations.

Acknowledgements.

This work was supported by the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq) and the Fundação de Amparo à Pesquisa do Estado São Paulo
(FAPESP). KNC and ASV acknowledge support from the AvH Foundation.

Appendix A Attenuated Witness

We would like to obtain an explicit expression for W′ppt(T1,T2) in terms of the physical parameters
of the bipartite system (variances and correlations). We note that the procedure cannot be directly realized
by first bringing V′ (or V) to a standard form and then applying the attenuation, since local symplectic
operations S∈Sp(2,R)⊕Sp(2,R) normally do not commute with the attenuation operation,
L(SVST)≠SL(V)ST(12); (28). Consequently, invariant quantities
under global and local symplectic transformations are not necessarily conserved by attenuations, such as the
global and local purities. On the other hand, SL(V)ST=L(SVST) is satisfied only if
SST=I, i.e. S must be a local phase space rotation, S∈SO(2,R)⊕SO(2,R).
Therefore, a criterion for entanglement robustness should depend solely on local rotational invariants.

We derive the explicit behavior of the witness W′ppt under attenuation.
Writing the PPT separability criterion in terms of the symplectic
invariants (6), we obtain

Wppt

=

1+detV+2detC−∑j=1,2detAj,

(47)

detV

=

detA1detA2+detC2−Λ4,

(48)

Λ4

=

tr(A1JCJA2JCTJ).

(49)

After attenuation, the matrices A1, A2, and C become

C′

=

√T1T2C,

(50)

A′i

=

Ti(Ai−I)+I,

(51)

To derive Eq. (17), we express the symplectic invariants in terms of quantities presenting
similar behavior. Two such quantities are obtained from Eq. (5)
and Eq. (50),

det(V′−I)

=

T21T22det(V−I),

(52)

detC′

=

T1T2detC.

(53)

Since for any 2×2 matrix M the following expressions are
valid,

det(M−I)

=

detM−trM+1,

(54)

tr(M−I)

=

tr(M)−2,

(55)

one obtains

ϖ′j−σ′j

=

T2j(ϖj−σj),

(56)

σ′j

=

Tjσj,

(57)

where σi=trAi−2, and ϖi=detAi−1
is the deviation from a pure state (impurity), which is zero for a pure state
and positive for any mixed state.

Applying Eq. (54) to det(V−I), we find quantities
which scale polynomially on the beam attenuations,

detV

=

det(V−I)+η,

(58)

η

=

σ1(ϖ2−σ2)+σ2(ϖ1−σ1)+σ1σ2

(59)

+

det(A1)+det(A2)+Λ1+Λ2−ΛC−1

Λ1

=

tr(CTJ(A1−I)JC),

Λ2

=

tr(CJ(A2−I)JCT),

ΛC

=

tr(CTC)

(60)

where the last three quantities scale as

Λ′1=T21T2Λ1

,

Λ′2=T1T22Λ2,

(61)

Λ′C

=

T1T2ΛC.

(62)

Substituting Eq. (58) in Eq. (47) and applying
the attenuation operation, we arrive at

W′ppt(T1,T2)

=

∑i,j=1,2Ti1Tj2Γij,with

(63)

Γ22

=

det(V−I)=det(V)−η,

Γ12

=

σ1(ϖ2−σ2)+Λ2,

Γ21

=

σ2(ϖ1−σ1)+Λ1,

Γ11

=

σ1σ2−ΛC+2det(C),

The function W′ppt describes the dynamics of all bipartite Gaussian states under losses.